Where do these hair artifacts come from?
I guess you are using curve segments as primitives for your hair. In this case Blender interpolates the segments between the simulated hair particles using cardinal splines. However it can only approximate the spline using linear, tube-like segments.
It looks like Blender does not close the gaps between the tubes properly. You can reduce the effect by increasing the Curve subdivisions in the Cycles Hair Rendering settings:



This will however hurt performance since more geometry is created.

Here is a comparison of Curve subdivisions set to 2 (top) versus Curve subdivisions set to 4 (bottom):




In the bottom image the white lines are gone, however you can still see that the curves are only approximated by straight tubular pieces (see the edge of the hair in the bottom left corner and the shading of the hairs in the bottom right corner).